Cryocoolers are often used to cool various components to extremely low temperatures. For example, cryocoolers can be used to cool focal plane arrays in different types of imaging systems. It is often necessary or desirable to secure certain components of a cryocooler in fixed positions relative to other components of the cryocooler. This may be needed, for example, to ensure proper operation of the cryocooler or to reduce disturbances in the cryocooler or in an overall system. One approach to securing components of a cryocooler involves the use of flexure bearings that connect moving mechanisms of the cryocooler to a support structure. A conventional flexure bearing includes arms arranged in a spiral pattern, where the arms extend between a moving mechanism of the cryocooler and the support structure.